Willmore submanifolds of the Möbius space and a Bernstein-type theorem
After calculating the structure equations and transformation formulae for Darboux frames of an immersion \(f:M^ m\to S^ n\) into the Möbius space \(S^ n\subset {\mathbb{R}}P^{n-1}\), the authors give a generalization of the Willmore functional for such immersions. Critical immersions of this functional are called ``Willmore submanifolds'', the Euler-Lagrange equation for the corresponding variational problem is derived. The hyperbolic conformal Gauß map, taking values in the conformal Grassmannian and a functional corresponding to the functional are derived. Using this framework a Bernstein-type theorem is formulated: If \(f:M^{n-1}\to S^ n\) \((n\neq 4,6)\) is a compact, oriented, connected Willmore hypersurface satisfying some additional condition with hyperbolic Gauß map \(\gamma_ f\) such that there exists a vector \(a\in {\mathbb{R}}_ 1^{n+2}\) with \(\langle\gamma_ f,a\rangle\neq 0\) on all \(M\), then \(f(M)\) is totally umbilic. Here the conformal Grassmannian is identified with the Lorentz sphere in Minkowski space \({\mathbb{R}}_ 1^{n+2}\), which is the parameter space of \({\mathbb{R}}P^{n+1}\).
scientific article; zbMATH DE number 510023